Abstract

We analyze the ability of nonlinear image restoration to remove interference artifacts in microscopes that enlarge the axial optical bandwidth through coherent counterpropagating waves. We calculate the images of an elaborate test object as produced by confocal, standing-wave, incoherent illumination interference image interference, and 4Pi confocal microscopes, and we subsequently investigate the extent to which the initial object can be restored by the information allowed by their optical transfer function. We find that nonlinear restoration is successful only if the transfer function is sufficiently contiguous and has amplitudes well above the noise level, as is mostly the case in a two-photon excitation 4Pi confocal microscope.

© 2001 Optical Society of America

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References

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  1. F. Lanni, Applications of Fluorescence in the Biomedical Sciences, 1st ed. (Liss, New York, 1986).
  2. B. Bailey, D. L. Farkas, D. L. Taylor, F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature (London) 366, 44–48 (1993).
    [CrossRef]
  3. S. W. Hell, “Double-scanning microscope,” European patent0491289 (December18, 1990).
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    [CrossRef]
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    [CrossRef]
  6. M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “I5M: 3D widefield light microscopy with better than 100 nm axial resolution,” J. Microsc. (Oxford) 195, 10–16 (1999).
    [CrossRef]
  7. M. Nagorni, S. W. Hell, “Coherent use of opposing lenses for axial resolution increase in fluorescence microscopy. I. Comparative study of concepts,” J. Opt. Soc. Am. A 18, 36–48 (2001).
    [CrossRef]
  8. V. Krishnamurthi, B. Bailey, F. Lanni, “Image processing in 3-D standing wave fluorescence microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 18–25 (1996).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. D. L. Snyder, T. J. Schutz, J. A. O’Sullivan, “Deblurring subject to nonnegative constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
    [CrossRef]
  16. G. M. P. Van Kempen, L. J. Van Vliet, P. J. Verveer, H. T. M. van der Voort, “A quantitative comparison of image restoration methods for confocal microscopy,” J. Microsc. (Oxford) 185, 354–365 (1997).
    [CrossRef]
  17. J.-A. Conchello, J. G. McNally, “Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 199–208 (1996).
    [CrossRef]
  18. I. Csiszar, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” Ann. Stat. 19, 2032–2066 (1991).
    [CrossRef]
  19. M. Schrader, K. Bahlmann, G. Giese, S. W. Hell, “4Pi-confocal imaging in fixed biological specimens,” Biophys. J. 75, 1659–1668 (1998).
    [CrossRef] [PubMed]

2001 (1)

1999 (1)

M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “I5M: 3D widefield light microscopy with better than 100 nm axial resolution,” J. Microsc. (Oxford) 195, 10–16 (1999).
[CrossRef]

1998 (1)

M. Schrader, K. Bahlmann, G. Giese, S. W. Hell, “4Pi-confocal imaging in fixed biological specimens,” Biophys. J. 75, 1659–1668 (1998).
[CrossRef] [PubMed]

1997 (1)

G. M. P. Van Kempen, L. J. Van Vliet, P. J. Verveer, H. T. M. van der Voort, “A quantitative comparison of image restoration methods for confocal microscopy,” J. Microsc. (Oxford) 185, 354–365 (1997).
[CrossRef]

1994 (1)

1993 (1)

B. Bailey, D. L. Farkas, D. L. Taylor, F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature (London) 366, 44–48 (1993).
[CrossRef]

1992 (2)

S. Hell, E. H. K. Stelzer, “Properties of a 4Pi-confocal fluorescence microscope,” J. Opt. Soc. Am. A 9, 2159–2166 (1992).
[CrossRef]

D. L. Snyder, T. J. Schutz, J. A. O’Sullivan, “Deblurring subject to nonnegative constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

1991 (1)

I. Csiszar, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” Ann. Stat. 19, 2032–2066 (1991).
[CrossRef]

1989 (1)

1988 (1)

1974 (1)

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

1972 (2)

Agard, D. A.

M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “I5M: 3D widefield light microscopy with better than 100 nm axial resolution,” J. Microsc. (Oxford) 195, 10–16 (1999).
[CrossRef]

M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Sevenfold improvement of axial resolution in 3D widefield microscopy using two objective lenses,” in Three-Dimensional Microscopy: Image Acquisition and Processing II, T. Wilson, C. J. Cogswell, eds., Proc. SPIE2412, 147–156 (1995).
[CrossRef]

Bahlmann, K.

M. Schrader, K. Bahlmann, G. Giese, S. W. Hell, “4Pi-confocal imaging in fixed biological specimens,” Biophys. J. 75, 1659–1668 (1998).
[CrossRef] [PubMed]

Bailey, B.

B. Bailey, D. L. Farkas, D. L. Taylor, F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature (London) 366, 44–48 (1993).
[CrossRef]

V. Krishnamurthi, B. Bailey, F. Lanni, “Image processing in 3-D standing wave fluorescence microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 18–25 (1996).
[CrossRef]

Conchello, J.-A.

J.-A. Conchello, J. G. McNally, “Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 199–208 (1996).
[CrossRef]

Csiszar, I.

I. Csiszar, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” Ann. Stat. 19, 2032–2066 (1991).
[CrossRef]

Farkas, D. L.

B. Bailey, D. L. Farkas, D. L. Taylor, F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature (London) 366, 44–48 (1993).
[CrossRef]

Frieden, B. R.

Giese, G.

M. Schrader, K. Bahlmann, G. Giese, S. W. Hell, “4Pi-confocal imaging in fixed biological specimens,” Biophys. J. 75, 1659–1668 (1998).
[CrossRef] [PubMed]

Gu, M.

Gustafsson, M. G. L.

M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “I5M: 3D widefield light microscopy with better than 100 nm axial resolution,” J. Microsc. (Oxford) 195, 10–16 (1999).
[CrossRef]

M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Sevenfold improvement of axial resolution in 3D widefield microscopy using two objective lenses,” in Three-Dimensional Microscopy: Image Acquisition and Processing II, T. Wilson, C. J. Cogswell, eds., Proc. SPIE2412, 147–156 (1995).
[CrossRef]

Hell, S.

Hell, S. W.

M. Nagorni, S. W. Hell, “Coherent use of opposing lenses for axial resolution increase in fluorescence microscopy. I. Comparative study of concepts,” J. Opt. Soc. Am. A 18, 36–48 (2001).
[CrossRef]

M. Schrader, K. Bahlmann, G. Giese, S. W. Hell, “4Pi-confocal imaging in fixed biological specimens,” Biophys. J. 75, 1659–1668 (1998).
[CrossRef] [PubMed]

S. W. Hell, “Double-scanning microscope,” European patent0491289 (December18, 1990).

Holmes, T. J.

Krishnamurthi, V.

V. Krishnamurthi, B. Bailey, F. Lanni, “Image processing in 3-D standing wave fluorescence microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 18–25 (1996).
[CrossRef]

Lanni, F.

B. Bailey, D. L. Farkas, D. L. Taylor, F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature (London) 366, 44–48 (1993).
[CrossRef]

F. Lanni, Applications of Fluorescence in the Biomedical Sciences, 1st ed. (Liss, New York, 1986).

V. Krishnamurthi, B. Bailey, F. Lanni, “Image processing in 3-D standing wave fluorescence microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 18–25 (1996).
[CrossRef]

Lucy, L. B.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

McNally, J. G.

J.-A. Conchello, J. G. McNally, “Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 199–208 (1996).
[CrossRef]

Nagorni, M.

O’Sullivan, J. A.

D. L. Snyder, T. J. Schutz, J. A. O’Sullivan, “Deblurring subject to nonnegative constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

Richardson, W. H.

Schrader, M.

M. Schrader, K. Bahlmann, G. Giese, S. W. Hell, “4Pi-confocal imaging in fixed biological specimens,” Biophys. J. 75, 1659–1668 (1998).
[CrossRef] [PubMed]

Schutz, T. J.

D. L. Snyder, T. J. Schutz, J. A. O’Sullivan, “Deblurring subject to nonnegative constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

Sedat, J. W.

M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “I5M: 3D widefield light microscopy with better than 100 nm axial resolution,” J. Microsc. (Oxford) 195, 10–16 (1999).
[CrossRef]

M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Sevenfold improvement of axial resolution in 3D widefield microscopy using two objective lenses,” in Three-Dimensional Microscopy: Image Acquisition and Processing II, T. Wilson, C. J. Cogswell, eds., Proc. SPIE2412, 147–156 (1995).
[CrossRef]

Sheppard, C. J. R.

Snyder, D. L.

D. L. Snyder, T. J. Schutz, J. A. O’Sullivan, “Deblurring subject to nonnegative constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

Stelzer, E. H. K.

Taylor, D. L.

B. Bailey, D. L. Farkas, D. L. Taylor, F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature (London) 366, 44–48 (1993).
[CrossRef]

van der Voort, H. T. M.

G. M. P. Van Kempen, L. J. Van Vliet, P. J. Verveer, H. T. M. van der Voort, “A quantitative comparison of image restoration methods for confocal microscopy,” J. Microsc. (Oxford) 185, 354–365 (1997).
[CrossRef]

Van Kempen, G. M. P.

G. M. P. Van Kempen, L. J. Van Vliet, P. J. Verveer, H. T. M. van der Voort, “A quantitative comparison of image restoration methods for confocal microscopy,” J. Microsc. (Oxford) 185, 354–365 (1997).
[CrossRef]

Van Vliet, L. J.

G. M. P. Van Kempen, L. J. Van Vliet, P. J. Verveer, H. T. M. van der Voort, “A quantitative comparison of image restoration methods for confocal microscopy,” J. Microsc. (Oxford) 185, 354–365 (1997).
[CrossRef]

Verveer, P. J.

G. M. P. Van Kempen, L. J. Van Vliet, P. J. Verveer, H. T. M. van der Voort, “A quantitative comparison of image restoration methods for confocal microscopy,” J. Microsc. (Oxford) 185, 354–365 (1997).
[CrossRef]

Ann. Stat. (1)

I. Csiszar, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” Ann. Stat. 19, 2032–2066 (1991).
[CrossRef]

Astron. J. (1)

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

Biophys. J. (1)

M. Schrader, K. Bahlmann, G. Giese, S. W. Hell, “4Pi-confocal imaging in fixed biological specimens,” Biophys. J. 75, 1659–1668 (1998).
[CrossRef] [PubMed]

IEEE Trans. Signal Process. (1)

D. L. Snyder, T. J. Schutz, J. A. O’Sullivan, “Deblurring subject to nonnegative constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

J. Microsc. (Oxford) (2)

G. M. P. Van Kempen, L. J. Van Vliet, P. J. Verveer, H. T. M. van der Voort, “A quantitative comparison of image restoration methods for confocal microscopy,” J. Microsc. (Oxford) 185, 354–365 (1997).
[CrossRef]

M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “I5M: 3D widefield light microscopy with better than 100 nm axial resolution,” J. Microsc. (Oxford) 195, 10–16 (1999).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Nature (London) (1)

B. Bailey, D. L. Farkas, D. L. Taylor, F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature (London) 366, 44–48 (1993).
[CrossRef]

Other (5)

S. W. Hell, “Double-scanning microscope,” European patent0491289 (December18, 1990).

M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Sevenfold improvement of axial resolution in 3D widefield microscopy using two objective lenses,” in Three-Dimensional Microscopy: Image Acquisition and Processing II, T. Wilson, C. J. Cogswell, eds., Proc. SPIE2412, 147–156 (1995).
[CrossRef]

F. Lanni, Applications of Fluorescence in the Biomedical Sciences, 1st ed. (Liss, New York, 1986).

V. Krishnamurthi, B. Bailey, F. Lanni, “Image processing in 3-D standing wave fluorescence microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 18–25 (1996).
[CrossRef]

J.-A. Conchello, J. G. McNally, “Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 199–208 (1996).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Test object: (a) axial section (XZ) encompassing the optical axis Z, designed to contain all axial distances of interest, and (b) YZ and XY sections of the 3D test object.

Fig. 2
Fig. 2

Raw image data of the object in Fig. 1 calculated for the confocal microscope, the SWM, the I5M, the 4Pi type C microscope, the two-photon 4Pi type A microscope, and the two-photon 4Pi type C microscope (left column), shown in the same section as that in Fig. 1(a). The data were normalized to a maximum of 100 counts before a Poisson noise filter was applied. Results of the RL restoration are given in the right column. The number of iterations is indicated in the lower right corner. Observe the structure of the raw image data and the varying degree of success of the restoration.

Fig. 3
Fig. 3

Calculated I5M raw data (left) and their RL restored counterparts (right). To account for a potentially higher signal in this microscope, we assumed an average maximum of 10,000 counts, which is 100 times higher than that in Fig. 2. Whereas in (a) the restored result exhibits artifacts, the restoration shown in (b) is successful, because the imaged object section (upper right inset) does not contain unfavorable spatial frequencies.

Fig. 4
Fig. 4

Comparison of 4Pi confocal type C images obtained by restoring with λ=5×10-6 (left), which is slightly too high, and λ=5×10-7 (right), which is too low. The profiles reveal that the residual lobe artifacts are not significantly affected. The position of the profiles in the image is indicated by the arrows. The two highest peaks correspond to the line object and the sphere; the two weaker peaks are artifacts.

Fig. 5
Fig. 5

Calculated raw data (left) of the 4Pi confocal type C, two-photon 4Pi confocal type A, and two-photon 4Pi confocal type C microscopes assuming a pinhole size of 87% of the backprojected fluorescence Airy disk. The comparison with the object data in Fig. 1 discloses a remarkable performance of the combination of two-photon 4Pi confocal microscopy with image restoration.

Equations (3)

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fˆm,nk+1=fˆm,nkijgi,jrshr,s fˆi-r,j-sk hi-m, j-n.
fˆregk+1=-1+(1+2λfˆ k+1)1/2λ,
I(f, fˆ )=ijfi, jlnfi, jf^i, j-(fi, j-f^i, j).

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